Estimates for $L$-functions in the critical strip under GRH with effective applications

Abstract

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log \mathcal{L}(s)$ and ${\mathcal{L}}^{\prime }(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood's well known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application we provide conditional and effective estimate for the Mertens function.

20 pages